1,023 research outputs found
Distributed Symmetry Breaking in Hypergraphs
Fundamental local symmetry breaking problems such as Maximal Independent Set
(MIS) and coloring have been recognized as important by the community, and
studied extensively in (standard) graphs. In particular, fast (i.e.,
logarithmic run time) randomized algorithms are well-established for MIS and
-coloring in both the LOCAL and CONGEST distributed computing
models. On the other hand, comparatively much less is known on the complexity
of distributed symmetry breaking in {\em hypergraphs}. In particular, a key
question is whether a fast (randomized) algorithm for MIS exists for
hypergraphs.
In this paper, we study the distributed complexity of symmetry breaking in
hypergraphs by presenting distributed randomized algorithms for a variety of
fundamental problems under a natural distributed computing model for
hypergraphs. We first show that MIS in hypergraphs (of arbitrary dimension) can
be solved in rounds ( is the number of nodes of the
hypergraph) in the LOCAL model. We then present a key result of this paper ---
an -round hypergraph MIS algorithm in
the CONGEST model where is the maximum node degree of the hypergraph
and is any arbitrarily small constant.
To demonstrate the usefulness of hypergraph MIS, we present applications of
our hypergraph algorithm to solving problems in (standard) graphs. In
particular, the hypergraph MIS yields fast distributed algorithms for the {\em
balanced minimal dominating set} problem (left open in Harris et al. [ICALP
2013]) and the {\em minimal connected dominating set problem}. We also present
distributed algorithms for coloring, maximal matching, and maximal clique in
hypergraphs.Comment: Changes from the previous version: More references adde
On Computing Maximal Independent Sets of Hypergraphs in Parallel
Whether or not the problem of finding maximal independent sets (MIS) in
hypergraphs is in (R)NC is one of the fundamental problems in the theory of
parallel computing. Unlike the well-understood case of MIS in graphs, for the
hypergraph problem, our knowledge is quite limited despite considerable work.
It is known that the problem is in \emph{RNC} when the edges of the hypergraph
have constant size. For general hypergraphs with vertices and edges,
the fastest previously known algorithm works in time with
processors. In this paper we give an EREW PRAM algorithm
that works in time with processors on general
hypergraphs satisfying , where
and . Our algorithm is
based on a sampling idea that reduces the dimension of the hypergraph and
employs the algorithm for constant dimension hypergraphs as a subroutine
Incremental complexity of a bi-objective hypergraph transversal problem
The hypergraph transversal problem has been intensively studied, from both a
theoretical and a practical point of view. In particular , its incremental
complexity is known to be quasi-polynomial in general and polynomial for
bounded hypergraphs. Recent applications in computational biology however
require to solve a generalization of this problem, that we call bi-objective
transversal problem. The instance is in this case composed of a pair of
hypergraphs (A, B), and the aim is to find minimal sets which hit all the
hyperedges of A while intersecting a minimal set of hyperedges of B. In this
paper, we formalize this problem, link it to a problem on monotone boolean
-- formulae of depth 3 and study its incremental complexity
Efficient enumeration of solutions produced by closure operations
In this paper we address the problem of generating all elements obtained by
the saturation of an initial set by some operations. More precisely, we prove
that we can generate the closure of a boolean relation (a set of boolean
vectors) by polymorphisms with a polynomial delay. Therefore we can compute
with polynomial delay the closure of a family of sets by any set of "set
operations": union, intersection, symmetric difference, subsets, supersets
). To do so, we study the problem: for a set
of operations , decide whether an element belongs to the closure
by of a family of elements. In the boolean case, we prove that
is in P for any set of boolean operations
. When the input vectors are over a domain larger than two
elements, we prove that the generic enumeration method fails, since
is NP-hard for some . We also study the
problem of generating minimal or maximal elements of closures and prove that
some of them are related to well known enumeration problems such as the
enumeration of the circuits of a matroid or the enumeration of maximal
independent sets of a hypergraph. This article improves on previous works of
the same authors.Comment: 30 pages, 1 figure. Long version of the article arXiv:1509.05623 of
the same name which appeared in STACS 2016. Final version for DMTCS journa
Trace Spaces: an Efficient New Technique for State-Space Reduction
State-space reduction techniques, used primarily in model-checkers, all rely
on the idea that some actions are independent, hence could be taken in any
(respective) order while put in parallel, without changing the semantics. It is
thus not necessary to consider all execution paths in the interleaving
semantics of a concurrent program, but rather some equivalence classes. The
purpose of this paper is to describe a new algorithm to compute such
equivalence classes, and a representative per class, which is based on ideas
originating in algebraic topology. We introduce a geometric semantics of
concurrent languages, where programs are interpreted as directed topological
spaces, and study its properties in order to devise an algorithm for computing
dihomotopy classes of execution paths. In particular, our algorithm is able to
compute a control-flow graph for concurrent programs, possibly containing
loops, which is "as reduced as possible" in the sense that it generates traces
modulo equivalence. A preliminary implementation was achieved, showing
promising results towards efficient methods to analyze concurrent programs,
with very promising results compared to partial-order reduction techniques
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